The Probability Calculator is a free online tool that calculates the probability of a single event based on the provided number of events. It can also determine the probabilities of independent events for the following eight scenarios:
- A and B both occurring: P(A ∩ B)
- A or B both occurring: P(A ∪ B)
- A Not occurring: P(A')
- B Not occurring: P(B')
- A or B occurs but Not both: P(A Δ B)
- Neither A nor B occurring: P((A ∪ B)')
- A occurring but not B: P(A but NOT B)
- B occurring but not A: P (B but NOT A).
An option is also available to see all of the above as a whole.
What is Probability?
Probability is the measure of the likelihood or chance of an event happening. Probability values range from 0 to 1; where 0 denotes the event is impossible and 1 shows the event is certain. It quantifies the uncertainty of events and makes predictions about their occurrence.
Probability can be written in percentages, fractions, or decimals. For example, the probability of rolling a 6 on a standard die is 1/6 which is equivalent to 16.67% or 0.1667.
Probability Formulas
Probability has two types of formulas, one for the single event and the other for the independent event.
The formula for Single Event Probability
Single-event probability is the likelihood of a specific outcome occurring in a single trial.
- Probability of an Event (E) Formula
The formula for finding the probability of a single event (E) is given below.
Probability of Event E = Number of Events Occurred / Number of Possible Outcomes
P(E) = n(E) / n(T)
To calculate the probability P(E) as a percentage, multiply the probability P(E) by 100.
P(E)% = P(E) x 100
- Complement Rule
Formula to calculate the percentage of the complement of the event E.
P(E’)% = [1 – P(E)] x 100
Probability Formulas for Independent Events
Two events are considered independent when the occurrence of one event does not affect the probability of another event happening. Use the following formulas to find probability of independent events.
- A and B both occurring: P(A ∩ B)
P(A ∩ B) = P(A) × P(B)
- A or B both occurring: P(A ∪ B)
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
- A Not occurring: P(A')
P(A') = 1 - P(A)
- B Not occurring: P(B')
P(B') = 1 - P(B)
- A or B occurs but Not both: P(A Δ B)
P(A Δ B)=P(A)+P(B)−2×P(A∩B)
- Neither A nor B occurring: P((A ∪ B)')
P((A ∪ B)') = 1 - P(A ∪ B)
- A occurring but not B: P(A but NOT B)
P(A but NOT B) = P(A) - P(A ∩ B)
- B occurring but Not A: P(B but NOT A).
P(B but NOT A) = P(B) - P(A ∩ B)
Types of Probability
Probability is defined as the measure of the likelihood or ranges between 0 and 1. It tells us the probability of any event occurring. Sometimes the likelihood of any event occurring is dependent on many other factors due to this, probability can be further divided in further three other probabilities. Each probability has its specialty and specific notation. These types such as:
1. Marginal probability
This probability comes if the likelihood of any event occurring without affecting the other events. It can be represented by the P (A) or read as the probability (P) of an event (A) occurring. For example: Draw the number three by rolling the dice for the first time this is the marginal probability because it is not affected by any factors.
2. Joint Probability
It is the likelihood of two events occurring at the same time that can be represented by the P (A and B) or P (A ∩ B). It can be read as the probability of A and B both occurring at the same time.
3. Conditional Probability
If two events independently occur without affecting by the first event that does not affect the likelihood of the occurrence of the second event. This probability occurs when a different event already occurs and the vertical line is used to represent condition probability occurring between the two events.
Properties of Probability
Probability is the fundamental concept that plays an exclusive role in mathematics and statistics to determine the possibility of the occurrence of any event. Some properties of probability are discussed below.
- The probability of any event can be calculated by taking the ratio of favorable outcomes to the number of possible outcomes of any event. P(E) = n(E)/n(t)
- The probability of the occurrence of any event is a non-negative number. If A is the event then P(A) ≥ 0.
- If any event is occurred then the maximum probability of this event is 1.
- If an event that cannot occur in all tries is known as an impossible event and its probability is 0.
- The probability of any event is always a positive number and always relies between 0 and 1. The probability of an event “E” is mathematically denoted as, 0 ≤ P (E) ≤ 1.
- The sum of the probabilities of different events occurring at the same time is equal to 1.
- The probability of any event that does not occur, then its probability can be found by taking the difference of the probability of occurring event (A) from “1” known as the complement probability of an event or represented by A’ or Ac. Mathematically, P(A’) = 1 – P(A).
- The sum of the probability of any event and the complement of the probability of this event is always “1”. Such as “P(A) + P(Ac) = 1”.
Probability Rules
Five basic rules of probability help to solve the mathematical problem and understand the behavior of the occurred random events.
- Addition Rule of Probability
If A and B are the two events then the union of the two events is found by using the addition rule.
- If A and B are not mutually exclusive then, P (A∪B) = P(A)+P(B)−P(A∩B).
- If A and B are mutually exclusive (disjoint) then, P (A∪B) = P(A) + P(B) because P(A∩B) = ϕ.
- Multiplication Rule of Probability
This rule is used to find probability of the intersection of the two events independent or not independent of one another. If A & B are two events then,
- If A and B are not independent then: P (A∩B) = P(A)⋅P(B∣A) = P(B)⋅ P(A∣B).
- If A and B are independent then, P (A∩B) = P(A)⋅P(B).
- Conditional Probability Rule
This rule is used to find the probability of one event with respect to another event that occurred already. Let A and B be two events while the event “B” occurred already then the condition probability of A by B is found with the below formula.
P (A∣B) = P (A∩B)/P(B), Always, P(B)>0
- Complement Rule of Probability
This rule helps to calculate the probability of the complement of the given event. If we know the probability of the event “A” then its complement probability is found by this formula.
P (Ac) = P(A’) = 1 – P(A)
- Total probability rule
This rule is used to show the total sum of the probability of one event under the condition of the other event that already occurred. If B events “B1, B2, B3, and … Bn” is already occurred then the sum of the probability A can be written as:
P(A)=∑i=1n P(A∣Bi) P(Bi),
Where “B1, B2, B3, and … Bn” are mutually exclusive events.
How to Calculate Probability?
Let’s consider some examples to learn how to find probability of single events and independent events.
Example 1: (For Single Event Probability)
A fair die is rolled. What is the probability that it shows the even numbers?
Solution:
Step 1: Determine the number of the Possible Outcomes.
A fair six-sided die has six faces so n(T) = 6
Step 2: Determine the number of events that occurred n(E)
The even numbers on a single die are 2, 4, and 6.
n(E) = 3
Step 3: Apply the Probability Formula.
P(E) = n(E) / n(T)
Step 4: Substitute Values.
P(E) = 3/6
P(E) = 0.5 or 50%
So the probability of rolling an even number on the fair six-sided die is 0.5 or 50%.
Example 2: (For Independent Event Probability)
Two students A and B can solve 60% and 75% of the problems from the exercise, respectively. Find the probability that either A or B can solve a problem chosen at random from that exercise.
Solution:
Step 1: Define the probabilities of each student solving a problem.
The probability that student A can solve a problem: P(A) = 60%
P(A) = 60/100 = 0.6
The probability that student B can solve a problem: P(B) = 75%
P(B) = 75/100 = 0.75
Step 2: Find the probability that both A and B can solve a problem (i.e. P(A∩B)).
Since the events are independent, (one student's ability does not affect the other). P(A∩B) can be found by the following formula:
P(A ∩ B) = P(A) × P(B)
= 0.6 × 0.75
P(A ∩ B) = 0.45
Step 3: Use the formula to calculate A or B both occurring.
∴ P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Step 3: Put the values into the above equation.
P(A ∪ B) = 0.6 + 0.75 - 0.45
P(A ∪ B) = 0.90
Thus, the probability that either Student A or Student B can solve a problem chosen at random from the exercise is 0.9 or 90%.
Frequently Asked Questions (FAQ’s)
What is the And/OR rule in probability?
If the two events are mutually exclusive calculate the probability that events use the OR rule, that is, P(A or B) = P (A∪B) = P(A) + P(B). However, if the events are independent and to find the probability of both events occurring simultaneously then use the AND rule, that is P (A and B) = P (A∩B) =P(A)⋅P(B).
How to calculate the probability of an event?
The probability of any event is calculated by taking the ratio of the number of favorable outcomes with the total number of outcomes performed for any experiment. Alternatively, Use our above Probability calculator to calculate the value of the probability of any event according to your data for dependent or independent events.
What is the probability of a possible and impossible event?
According to the probability properties, the probability of the possible event is “1” and the impossible event is “0”.
If A and B are mutually exclusive events then the probability of P (A⋂B)?
The given events are mutually exclusive then the P (A⋂B) = ϕ.
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